Coupled Proca theories: Green-hyperbolicity, quantization and applications to polarization measurement
The Proca field describes a massive relativistic spin-$1$ particle and was originally formulated in Minkowski spacetime. Here we consider a variety of generalizations in globally hyperbolic spacetimes, including couplings between a number of Proca fields via a mass-matrix, the charged Proca field with arbitrary magnetic moment in an arbitrary external electromagnetic field, and a Proca-Klein-Gordon theory with a spacetime-dependent bilinear coupling. The equations are analysed using a general auxiliary field method, introduced here, which provides practical criteria for showing that a given operator is (semi)-Green-hyperbolic. The method goes beyond what is achieved in existing analyses of deformed equations, which for example place restrictions on the magnetic moment and electromagnetic potential that can be coupled to a Proca field. The theories considered can be quantized following a common pattern and all the examples treated in this work admit Hadamard states on any globally hyperbolic spacetime. As an application, the Proca-Klein-Gordon system is used to develop a measurement scheme sensitive to the Proca polarization, using a Klein-Gordon field as the probe. For a suitable family of $n$-particle Proca states, the leading-order probe response accords with Malus' law, confirming that this system acts as a polarization-sensitive detector.